Heat Exchanger Sizing: LMTD and Effectiveness-NTU Methods

The log-mean temperature difference is the driving-force average of the two terminal temperature differences across the exchanger. For a pure counterflow arrangement the LMTD is the highest achievable for a given temperature program, which is why counterflow is the thermodynamically preferred layout: less area is needed for the same duty. The required heat-transfer area follows directly from the overall heat-transfer coefficient U and the heat duty Q.

For any arrangement other than pure counterflow (parallel flow, 1-2 shell-and-tube, or cross-flow), the counterflow LMTD is multiplied by a correction factor F ≤ 1.0 that accounts for the departure from ideal counterflow. When F drops below 0.75 the required area becomes very sensitive to small changes in the temperature program, and adding shell passes or switching to counterflow should be considered. The LMTD engine in the calculator rejects temperature programs where a terminal ΔT ≤ 0 — a second-law violation — rather than clamping it to a small value and reporting a spuriously large area.

For a 1-2 shell-and-tube exchanger (one shell pass, two tube passes) the correction factor F is given by the Bowman-Mueller-Nagle (1940) closed form, which is a function of the dimensionless temperature parameters P (cold-stream thermal effectiveness) and R (heat-capacity rate ratio). P and R are defined from the four terminal temperatures alone, without knowing the flow rates. The Bowman formula returns no valid result when the temperature program (P, R) lies outside the physical single-shell-pass domain — the calculator reports this as a domain violation and suggests adding shell passes rather than silently returning a floored constant.

For a single-pass cross-flow exchanger with both fluids unmixed, there is no simple closed-form F expression analogous to the Bowman formula. The calculator uses the effectiveness-NTU inversion method (Incropera & DeWitt §11.4): it computes the exchanger effectiveness ε and capacity ratio C* from (P, R), finds NTU_counterflow and NTU_crossflow for the same ε and C*, and returns F = NTU_counterflow / NTU_crossflow. This matches the published F-correction charts (TEMA / ESDU 86018). For pure counterflow F = 1.0 exactly — no calculation needed.

The effectiveness-NTU method is used for rating: you know the exchanger (area A, overall U) and the stream conditions (inlet temperatures and flow rates), and want to find the heat duty and outlet temperatures without guessing outlet temperatures and iterating. The number of transfer units NTU = U·A / C_min is a dimensionless measure of exchanger size relative to the weaker stream. The capacity ratio C* = C_min / C_max (0 ≤ C* ≤ 1) describes the stream balance; C* = 0 represents a condensing or evaporating stream (infinite capacity rate on one side).

The effectiveness ε is the ratio of actual to maximum possible duty. Q_max is limited by the stream with the smaller heat-capacity rate C_min = min(ṁ·c_p), because that stream undergoes the largest temperature change. Once ε is found from the ε-NTU correlation for the flow arrangement, the actual duty Q_actual = ε · Q_max, and the outlet temperatures follow from Q_actual / C_h and Q_actual / C_c. The counterflow and 1-2 shell-and-tube ε-NTU expressions have compact closed forms; the both-unmixed cross-flow expression (Incropera Eq. 11.32) is a series approximation used in the calculator.

A clean exchanger seldom stays clean in service. Fouling deposits on the heat-transfer surfaces add thermal resistance and reduce the effective overall coefficient below U_clean. The fouled coefficient U_fouled is found by adding the hot-side and cold-side fouling resistances from the TEMA fouling tables to the clean overall resistance 1/U_clean. Because A = Q / (U·F·LMTD), a lower U_fouled means a larger required area — the extra area A_margin = A_fouled − A_clean is the design allowance for fouling over the service life.

The cleanliness factor CF = U_fouled / U_clean (always ≤ 1) is a compact way to express the severity of the fouling allowance. A CF of 0.80 means the exchanger must be 25% larger than the clean-area calculation to maintain the design duty after fouling. TEMA fouling resistances for common service fluids are built into the calculator's fouling panel.